- #1

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(I used simple polar coordinates in two dimensions)

[itex]H(r,p_{r})= \frac{p^{2}_{r}}{2m}+V(r)[/itex]

[itex]H(\phi, p_{\phi})=\frac{p^{2}_{\phi}}{2mr^{2}}+V(\phi)[/itex]

Now I know that if you take the Poisson bracket with the Hamiltonian you just get the infinitesimal transformation in time right? So

[itex]\{r,H(r,p_{r})\}= \frac{p_{r}}{m}[/itex]

[itex]\{\phi,H(\phi,p_{\phi})\}= \frac{p_{\phi}}{mr^{2}}[/itex]

But what if I want to do an infinitesimal transformation of the [itex]r[/itex] and [itex]\phi[/itex] coordinates? I know that the generator of translations is just the momentum, and that the generator of rotations is angular momentum. How would I do that with the Poisson bracket in this case? And for example when I do an infinitesimal transformation with the radius [itex]r[/itex], what does that mean? Is it that the radius is infinitesimally transformed, or is it more like a global translation where the whole system is somehow translated? Similarly with the angle [itex]\phi[/itex], is it that the angle is locally changed, or is it that the "whole" system is rotated?